Understanding Conditional Statements, Converses, and Biconditionals in Geometry

1. Honors Geometry Unit 2 – day 4 Conditional Statements Converses Biconditionals

2. Warm Up What is the fourth point of plane XUR Name the intersection of planes QUV and QTX Are point U and S collinear?

3. Quiz

4. 2-1 Conditional Statements • Objectives • To recognize conditional statements • To write converses of conditional statements

5. If-Then Statements • “If it is Valentine’s Day, then it is February.” • Another name of an if-then statement is a conditional. • Parts of a Conditional: • Hypothesis (after “If”) • Conclusion (after “Then”) “If you are not completely satisfied, then your money will be refunded.” (hypothesis) (conclusion)

6. Identifying the Parts • Identify the hypothesis and the conclusion of this conditional statement: • If it is Halloween, then it is October • Hypothesis: It is Halloween • Conclusion: It is October

7. Writing a Conditional • Write each sentence as a conditional: • A rectangle has four right angles “If a figure is a rectangle, then it has four right angles.” • An integer that ends with 0 is divisible by 5 “If an integer ends with 0, then it is divisible by 5.”

8. Truth Value A conditional can have a truth value of true or false. To show that a conditional is true, you must show that every time the hypothesis is true, the conclusion is also true. To show that a conditional is false, you need to only find one counterexample

9. Example • Show that this conditional is false by finding a counterexample • “If it is February, then there are only 28 days in the month” • Finding one counterexample will show that this conditional is false • February 2012 is a counterexample because 2012 was a leap year and there were 29 days in February

10. Converses • The converse of a conditional switches the hypothesis and the conclusion • Example • Conditional: “If two lines intersect to form right angles, then they are perpendicular.” • Converse: “If two lines are perpendicular, then they intersect to form right angles.”

11. Example • Write the converse of the following conditional: • “If two lines are not parallel and do not intersect, then they are skew” • “If two lines are skew, then they are not parallel and do not intersect.”

12. Are all converses true? • Write the converse of the following true conditional statement. Then, determine its truth value. • Conditional: “If a figure is a square, then it has four sides” • Converse: “If a figure has four sides, then it is a square” • Is the converse true? • NO! A rectangle that is not a square is a counterexample!

13. Assessment Prompt • Write the converse of each conditional statement. Determine the truth value of the conditional and its converse. • If two lines do not intersect, then they are parallel • Converse: “If two lines are parallel, then they do not intersect.” • Conditional is false • Converse is true • If x = 2, then |x| = 2 • Converse: “If |x| = 2, then x = 2” • Conditional is true • Converse if false

14. 2-2 Biconditionals • Objectives • To write biconditionals

15. 2-2 Biconditionals When a conditional and its converse are true, you can combine them as a biconditional. This is a statement you get by connecting the conditional and its converse with the phrase if and only if (iff).

16. Example of a Biconditional • Conditional • If two angles have the same measure, then the angles are congruent. • True • Converse • If two angles are congruent, then the angles have the same measure. • True • Biconditional • Two angles have the same measure if and only if the angles are congruent.

17. Example • Consider this true conditional statement. Write its converse. If the converse is also true, combine them as a biconditional • If three points are collinear, then they lie on the same line. • If three points lie on the same line, then they are collinear. • Three points are collinear if and only if they lie on the same line.

18. Definitions A good definition is a statement that can help you identify or classify an object. A good definition can be written as a biconditional.

19. Example • Show that this definition of perpendicular lines is a good defintion and that it can be written as a biconditional • Definition: Perpendicular lines are two lines that intersect to form right angles. • Conditional: If two lines are perpendicular, then they intersect to form right angles. • Converse: If two lines intersect to form right angles, then they are perpendicular. • Biconditional: Two lines are perpendicular if and only if they intersect to form right angles.

20. Real World Examples • Are the following statements good definitions? Explain • An airplane is a vehicle that flies. • Can it be written as a biconditional? • NO! A helicopter is a counterexample because it also flies! • A triangle has sharp corners. • Can it be written as a biconditional? • NO! Squares have sharp corners. (Sharp is not a precise word)

21. Homework Worksheet Logic Quiz Monday!